An important problem in conventional beamforming (CBF) in radar or sonar using a set of sensors is to determine the shading coefficients (weights) to be applied at the output of each sensor for superior sidelobe reduction. Shading coefficients help to reduce the sidelobes at the expense of a slight increase in the main lobe widths. The approach is to maximize the energy over the main beam and minimize the energy over the sidelobes by holding the total energy to be constant. This is realized by superimposing a window of suitable arbitrary shape and width over the main beam. Ideally these shading coefficients should be robust enough so as not to generate significant performance degradation when one or more of the sensors become inoperative.
In this context, let {ak}k=1M represent the shading coefficients for a uniform linear array with M sensors as shown in FIG. 1a. FIG. 1a includes sensors 10, 14, and 18. FIG. 1a also includes shading weights 12, 16, and 20. Sensor 10 has a shading weight of a1, sensor 14 has a shading weight of a2, sensor 18 has a shading weight of aM. Withω=π sin θ  (1)representing the normalized look-direction, where θ represents the physical look direction (or arrival angle) from the broadside of the array of sensors. From 10, 14, and 18, the array factor and the array gain pattern are given by
                              A          ⁡                      (            ω            )                          =                              ∑                          k              =              1                        M                    ⁢                                    a              k                        ⁢                          ⅇ                                                -                  j                                ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                ω                                      ⁢                                                  ⁢            and                                              (        2        )                                          G          ⁡                      (            ω            )                          =                                                                          A                ⁡                                  (                  ω                  )                                                                    2                    =                                                                                    ∑                                      k                    =                    1                                    M                                ⁢                                                      a                    k                                    ⁢                                      a                    k                                    ⁢                                      ⅇ                                                                  -                        j                                            ⁢                                                                                          ⁢                      k                      ⁢                                                                                          ⁢                      ω                                                                                                          2                                              (        3        )            respectively. A typical array gain pattern or the prior art is as shown in FIG. 1b. with a dominant main beam 102 having a width W1, surrounded by sidelobes 104a and 104b. The goal of the shading weights such as a1, a2, and aM is to enhance the main lobe 102 while maintaining the sidelobes 104a and 104b to be as uniform as possible. Further the sidelobes 104a and 104b should be robust enough so as to maintain similar low sidelobe structure when one or more of the sensors, such as 10, 14, or 18 of FIG. 1a become inactive. With equal shading for an M element array, Eq. (3) gives rise to the standard gain pattern
                                                        G              1                        ⁡                          (              ω              )                                =                                    (                                                sin                  ⁡                                      (                                          M                      ⁢                                                                                          ⁢                                              ω                        /                        2                                                              )                                                                    sin                  ⁡                                      (                                          ω                      /                      2                                        )                                                              )                        2                          ,                            (        4        )            and with triangular shading weights
                              a          k                =                  {                                                                      k                  ,                                                                              k                  ≤                                      M                    /                    2                                                                                                                                            M                    -                    k                                    ,                                                                              k                  >                                      M                    /                    2                                                                                                          (        5        )            the gain pattern simplifies to
                                          G            2                    ⁡                      (            ω            )                          =                                            (                                                sin                  ⁢                                      {                                                                  (                                                  M                          +                          1                                                )                                            ⁢                                              ω                        /                        4                                                              }                                                                    sin                  ⁡                                      (                                          ω                      /                      2                                        )                                                              )                        4                    .                                    (        6        )            
Compared to (4), notice that the sidelobe levels in (6) have been reduced by a factor of two (in the dB scale) while the main beam width has gone up by an undesirable factor of two as well. Thus shading helps to reduce the sidelobe level. However, the weights in (5) are not particularly attractive since their dynamic range is quite large especially for large arrays. In this context, Dolph-Chebyshev shading weights are widely used in practice since for a given sidelobe level, they achieve the minimum transition band thereby maintaining an optimum main lobe width as specified in “Array Signal Processing,” by S. U. Pillai, Springer-Verlag, N.Y., 1989, and “Array Signal Processing: Concepts and Techniques,” by Don H. Johnson and Dan E. Dugeon, PTR Prentice-Hall, 1993.
Most of the shading schemes such as Dolph-Chebyshev, Kaiser windows, Blackman-Harris windows assume a uniformly placed linear array of sensors, such as sensors 10, 14, 18, the spacing being λ/2, and this assumption is often violated in practice when hydrophones, such as 10, 14, and 18 become inoperative. In such a situation, if for example the mth hydrophone becomes non-operational, the actual gain pattern in (3) becomes
                                          G            m                    ⁡                      (            ω            )                          =                                                                                                                ∑                                          k                      =                      1                                        M                                                        k                    ≠                    m                                                  ⁢                                                      a                    k                                    ⁢                                      ⅇ                                                                  -                        j                                            ⁢                                                                                          ⁢                      k                      ⁢                                                                                          ⁢                      ω                                                                                                          2                    .                                    (        7        )            
In general, the array gain pattern associated with missing sensor elements have much worse sidelobe levels compared to its original counterpart. In this context it is desirable to have a set of shading coefficients whose performance degradation with respect to sidelobe suppression is graceful under missing sensors or sensor failures.